Chapter 2
Experimental Procedure
The chapter starts with a brief description of the deposition technique used for layer growth.
Then the characterization tools used in this work are outlined. Particular emphasis is laid
on the analysis of Raman and X-ray diraction experiments which served as a standard
tool for phase identication and structural characterization. Potential and limitations of
the respective techniques with respect to thin lm characterization are discussed.
2.1
System and Process Design
In general chalcopyrite thin lms are deposited under non-stoichiometric conditions, where
the chalcogen is oered at over-stoichiometric concentration. However nal lm compositions lie in the immediate vicinity of the Cu2 S-In2 S3 pseudo-binary line (Figure 1.4). This
400 ◦C)
is due to the high chalcogen vapor pressure at the deposition temperature (>
that hinders the incorporation of additional chalcogen atoms into the lattice. Depending
∆m (Equation 1.1) of the elemental composition growth
Cu-poor (∆m < 0) and Cu-rich (∆m > 0) ones. As discussed
on the deviation of molecularity
processes are divided into
in Section 1.1.2 structural properties and the defect chemistry of the chalcopyrite phase
are greatly inuenced by the
∆m
parameter, independently of the specic lm deposition
method [33, 36, 38, 57]. Sophisticated coevaporation processes which are nowadays used for
absorber layer preparation of the most ecient chalcopyrite thin lm solar cells utilize this
feature of the chalcopyrite system as they switch between the Cu-rich regime (to provoke
large grain sizes and good structural properties) and the In-rich regime (to avoid secondary
phases in the nal layer) during lm growth. Such processes are generally referred to as
two-stage or three-stage processes.
Several deposition techniques have been reported for thin lm growth of CuInS2 , i.e. single
source evaporation [58], coevaporation from single sources [59], chemical vapor transport
[60], spray pyrolysis [61], and reactive sputtering [62]. However, up to now device grade
CuInS2 material has only been achieved by either coevaporation of the elements or by
two step processes were a precursor layer of metallic phases or binary suldes is converted
into a CuInS2 lm by a reactive annealing step in a sulfur containing atmosphere [33,
21
22
Experimental Procedure
63, 64]. The growth process investigated in this work follows the latter approach, i.e. the
sulfurization of metallic Cu-In and Cu-(In,Ga) stacks. In such a two step process, metal
deposition and chalcogen incorporation are separate steps and the precursor composition
mainly determines the molecularity. In the case of CuInS2 thin lm growth for photovoltaic
applications the most ecient heterojunction solar cells have been achieved on the basis
of Cu-rich prepared absorber layers [63, 65].
2.1.1 Process Design for CuInS2 Thin Film Solar Cells
The general structure and the interface band line up of a chalcopyrite heterojunction solar
cell was already discussed in Section 1.2.1. Figure 2.1 shows the schematic structure and
a typical SEM cross section of a CuInS2 solar cell prepared in this work. The structure
consists of a metallic back contact (Mo), the
p-type absorber layer (CuInS2 ) and the n-type
window layer (CdS + ZnO). The ZnO window layer is contacted by a Ni/Al grid. The
process sequence is schematically depicted in Figure 2.2. In the following the preparation
steps will be briey outlined.
NiAlfront contact
n+-ZnOwindow layer
ZnO
n-CdSbuffer layer
p-CuInS2absorber layer
CuInS2
Mo-back contact
Mo
float glas-substrate
1 µm
SEM image
Figure 2.1: Schematic structure and SEM image of a cross-section of a CuInS2 thin lm
solar cell showing the dierent layers which constitute the device. The SEM image has
been shaded to highlight the dierent layers of the structure (marked CdS layer in the
image is not to scale).
Back contact deposition
used as sample substrates.
Conventional oat glass substrates (thickness 2 mm) were
The molybdenum back contact (layer thickness 1 µm) was
400 ◦C).
deposited by e-gun evaporation onto heated substrates (≈
2.1. System and Process Design
back
contact
absorber
23
window
layer
layer
CTP
PVD
PVD
e-gun
deposition
of Mo
thermal
evaporation
of
Cu-In
reactive
annealing
in
sulfur vapor
RTP
PVD
thermal
evaporation
of
Cu-(In,Ga)
reactive
annealing
in
sulfur vapor
CBD
KCN etching
and
deposition
of CdS
Sputtering
RF magnetron
sputtering
of
ZnO/ZnO:Al
PVD
e-gun
deposition
of
Ni/Al grids
RTP
reactive
annealing
in
H2S/Ar
Figure 2.2: Process sequence used in this work for CuInS2 thin lm solar cell preparation.
(PVD = physical vapor deposition, CTP = conventional thermal processing, RTP = rapid
thermal processing, CBD = chemical bath deposition).
Precursor deposition
Metal precursor layers were sequentially deposited by physical
vapor deposition (PVD) from tungsten boats in a high-vacuum-system.
heating was applied.
No substrate
Ultrapure pellets (99.9999 %) of Cu, In and Ga served as source
materials. The deposited thickness was monitored in-situ by separate X-tal balances for
each element. The atomic composition of the precursor was determined by adjusting the
lm thickness of each precursor layer. In addition the atomic ratios were controlled exsitu by determining the amount of evaporated material. A precise determination of the
precursor stoichiometry was realized by diluting the entire metal stack in a solution of
Königswasser ). Then the
Inductively Coupled Plasma
concentrated HCl and concentrated HNO3 at a ratio of 3:1 (
elemental concentrations in the solution were determined by
Atomic Emission Spectroscopy
(AES-ICP). Samples were stored in dry-air before further
processing. The typical structure of a Cu-In-Ga precursor stack is depicted in Figure 2.3.
Reactive annealing
The metallic precursor lms were transformed to chalcopyrite ab∗
sorber layers by reactive annealing or sulfurization in an atmosphere containing either
sulfur vapor Sx or a H2 S/Ar mixture (5% H2 S). Three systems were available for this temper step, which mainly diered in the reactive atmosphere used for sulfurization and in
the substrate heater design.
∗
The terms reactive annealing and sulfurization will be used equally in the text. Both terms refer to
the transformation of a precursors into a chalcopyrite in a sulfur containing atmosphere.
24
Experimental Procedure
Ga
In
Cu
Mo
Figure 2.3: Structure of metal-
float glass
600
500
o
Tmax =567 C
500
o
Tmax =547 C
o
Tmax =500 C
o
400
Temperature ( C)
o
Substrate Temperature ( C)
lic Cu-In-Ga stack (precursor).
300
200
conventional process
100
0
10 20 30 40 50 60 70 80 90
Time (min)
(a)
Figure 2.4:
300
RTP : sulfur vapor
thermo couple
Pt 100 on Mo
Pt 100 on CIS
200
100
RTP
0
400
0
0
60
120 180
Time (s)
240
300
(b)
(a) Characteristic temperature prole of a RTP sulfurization process and
a conventional process.
(b) Temperature prole of RTP annealing step (Sx system) as
measured by a thermo couple, a platinum resistor at the Mo back layer and a platinum
resistor at the CuInS2 layer (redrawn from [66]).
2.1. System and Process Design
25
1.) Conventional Thermal Processing (CTP): In this system a reactive atmosphere was
◦
established from a source containing a liquid sulfur bath at T = 200 C. Process pressure
−3
mbar during sulfurization. Substrates were heated by an conventional
was around 10
◦
◦
resistor heater to temperatures around 500 C. Annealing times at 500 C were between 15
to 30 minutes.
2.)
Rapid Thermal Processing (RTP): Two RTP-systems were also used for precursor
sulfurization. Due to the low thermal mass of such a system (substrates are heated by a
UV-lamp eld) very short heat up and cool down ramps can be realized. Therefore the
RTP systems were also used for quenching experiments which allowed for an ex-situ investigation of the lm growth behavior (Chapter 3). Sulfur was supplied either by putting
sulfur powder next to the substrate which evaporates once the substrates are heated or
by a H2 S/Ar gas ow.
Process pressure was around 1 mbar in the sulfur vapor system
and 500 mbar in the H2 S/Ar system. Typical annealing times were as short as 5 minutes.
Figure 2.4 (a) shows characteristic temperature proles of a RTP sulfurization process and
a conventional process. Experimental values for the substrate temperature were found to
vary up to
≈ 50 K
depending on the specic sensor and/or on the probing point at the
sample or sample holder (Figure 2.4).
Substrate temperature values given in this work
are always as read on the meter, thus they might be the subject to a substantial absolute
error.
However, relative changes in temperature were found to be very reproducible in
all systems.
Since dierent devices were used for temperature control in the individual
systems substrate temperature values can not be compared directly between the dierent
systems.
Window layer deposition
The window layer consist of
n-type
CdS buer layer of
+
about 50 nm thickness, a n-type ZnO layer of 100 nm thickness, and a n -type ZnO:Al
layer. Before the window layers are deposited the secondary CuS phase which forms during reactive annealing is removed by selective etching in a KCN solution. The CdS buer
layer is deposited immediately after KCN etching by chemical bath deposition at a sub◦
strate temperature of 60 C. The ZnO layers are sequentially deposited by RF magnetron
sputtering from a ZnO and a ZnO:Al target.
evaporation.
Ni/Al front grids are deposited by e-gun
26
2.2
Experimental Procedure
Structural Characterization
Several characterization techniques have been employed for sample analysis, i.e. secondary
neutral mass spectrometry (elemental depth proling), micro-Raman spectroscopy and Xray diraction (phase analysis, structural analysis) and inductively coupled plasma atomic
emission spectroscopy (atomic sample composition). Potential advantages and limitations
of these techniques with respect to thin lm analysis will be discussed in the following. Information about lm roughness and morphological grain size were obtained from scanning
electron microscopy images (SEM). In addition a small number samples was forwarded to
transmission electron microscopy (TEM).
2.2.1 Depth Proling
In order to determine the elemental composition and depth distribution of Cu(In1−x Gax )S2
thin lms secondary neutral atom mass spectrometry (SNMS) was used. SNMS is very
similar to the widely used secondary ion mass spectrometry (SIMS), but here neutral
atoms, which are ionized separately, are used for mass spectrometry. Mass analysis in a
SIMS measurement is based on the fact that a certain fraction (less than 1 % [67]) of the
sputtered species is ionized.
This fraction, i.e. the ionization yield, diers for dierent
elements and compounds and is also sensitive to the structural properties of the sample.
As long as the yield is known the nal result can be corrected, however, especially in
polycrystalline samples the ionization yield might well be inuenced by local variations
of the micro-structure in the vicinity of the sputtered atom (matrix eect), by oxidized
surfaces and by changing surface compositions [68]. In SNMS, the uncertainty introduced
by these eects is circumvent by separating the sputtering and the ionizing process. Only
neutral atoms, which are ionized in a separate ionizer are used for analysis.
The sput-
tering yield, i.e. the fraction of eroded atoms per incident ion ux, is, in general, also
a function of the above mention sample specic parameters.
However, after some ten
nanometers surface depletion and sputtering yield of each atomic specimen in the sample
will approach an dynamic equilibrium, so that the composition of the sputtered secondary
atoms reects the sample composition; in other words the sputtering process self adjusts.
In a polycrystalline sample this might not be fully true as the sputtering yield can vary
across the sample volume, in particular if the sample exhibits phase boundaries or large
changes in structural properties (grain size, defect density etc.). When analyzing the experimental output it is dicult to dierentiate such changes in sputtering yield from pure
compositional changes.
Furthermore Cu-chalcopyrites haven proven to be very suscep-
tible to roughness development during Ar-sputtering [69]. Similar eects are known for
indium containing III-V compounds.
In the latter case the eect has been ascribed to
indium enrichment at the surface, which occurs due to the preferential sputtering of the
V-compound and surface diusion of indium [70]. As a result of the dierent etch rates of
indium and the surrounding In-V compound, the In-islands act as a shield for the underly-
2.2. Structural Characterization
27
ing layer leading to an increase in surface roughness with ongoing sputtering [70]. Sample
rotation [69] and reactive sputtering with N2 [70] have been suggested to overcome this
problem. In the latter case the suppression of In-segregation was tentatively assigned to
Indium nitride formation at the surface, which has roughly the same sputter yield as the
bulk material. Furthermore it is believed to protect the underlying regions from sputterdamage (important when using sample for PL or Raman) [71]. Another source of error is
radiation-enhanced diusion [68]. Sputter induced defects at the surface can migrate into
the bulk, thereby changing the original composition of the sample. According to Eicke [72]
Cu depth proles of Cu(In1−x Gax )Se2 were found to be very sensitive to sputtering conditions, which was related to the high mobility found for Cu in chalcopyrite lattices [73]. By
signicantly reducing the substrate temperature to e. g. liquid nitrogen temperature such
eects can be reduced [67]. Surface roughness might be another problem, which can have
detrimental eects on the depth resolution of the measurement, since the sputtering yield
depends on the angle between the incident ion beam and the sample surface. As a rule
◦
of thumb in many cases the yield is greatest at an angle of 45 . That means, assuming
normal incidence of the sputter species, a tilted surfaces of a crystalline grain will sputter
o faster than a surface parallel to the substrate. Hence, surface roughness is preserved
or even enhanced by the etching process. Again, the eect can be signicantly reduced by
rotating the sample during the process. However, although some of the discussed eects
can be minimized under certain experimental conditions, they have to be considered when
analyzing elemental depth proles from SIMS or SNMS experiments.
All SNMS measurements reported in this work have been carried out in a LHS10 system
with a SSM 200 (electron beam SNMS) at the Zentrum für Sonnenenergie- und WasserstoForschung Stuttgart (ZSW). The SNMS proles were collected under the following condi+
◦
tions. An focused Ar beam of E = 5 keV and I ≈ 1 µA under an incident angle of 60
(with respect to the sample normal) was used for sputtering. Two dierent sputter modes
have been used: 1.
ature, and 2.
stationary samples which were cooled down to liquid nitrogen temper-
rotated samples at room temperature. The sputtered species were ionized
by means of a cross-beam electron source (E = (5080) keV), whereby a electronic lens system assured that only neutral atoms could reach the ionizer. Then the ionized atoms were
passed through a quadrupol mass spectrometer and nally detected by a Cu-Be-dynode.
Secondary electrons emitted from the dyode were amplied by a photomultiplier and a
pre-amplier to a recordable signal. The raw data in counts per seconds was converted
to concentration proles by using experimentally determined sensitivity factors.
These
factors where based on system calibrations performed at Cu(Ga,In)(S,Se)2 single crystals
and thin lms of known composition.
There was a signicant variation in total sputter rate as a function of sample composition
and a drastic drop in sputter rate once the Mo-back contact layer was reached. Figure 2.5
shows the sum of the Cu, In, Ga, and S count rate, corrected by the respective sensitivity
factor. The eect of lm morphology and composition onto the sputter rate can clearly
be seen in the gure.
Without sample rotation (subgure (a.))
lms with larger grains
sputter o faster and the rate varies quite signicantly over time. The eect is substan-
28
Experimental Procedure
dgrain > 1 µm
stationary sample
2500
dgrain > 1 µm
rotated sample 1000
Tsubst. = RT 800
o
Tsubst. = -180 C
Counts
2000
600
1500
1000
500
0
400
dgrain < 1 µm
CuGaS2 / CuInS2
CuInS2 / CuGaS2
all plots
Cu(In,Ga)S2 / CuGaS2
0
2
4
Counts
3000
200
0
6
8
10
12
Sputter Time (1000 s)
0
10
20
30
40
50
60
Sputter Time (1000 s)
(a.)
(b.)
Figure 2.5: Sum of corrected count rate of Cu, In, Ga, and S versus sputter time indicating
sample dependent variations in sputter rate caused by (a) the grain sizes of the lm, and
(b) by the stacking sequence of dierent phase in the lm.
tially reduced, if the samples is rotated during sputtering (subgure (b.)). Since the data
in subgure (b) was collected at CuInS2 /CuGaS2 bilayers of similar grain size but dierent
stacking sequence the changes in total sputter rate have to be assigned to the changing
composition with depth. The plots indicate a higher sputter rate for polycrystalline CuInS2
when compared to CuGaS2 . The sputter rate of CuGaS2 seems to be by one fth smaller
compared to CuInS2 .
2.2.2 Micro-Raman Spectroscopy
Raman spectroscopy has become a widely used method for the analysis of semiconductor layers. Its non-destructive nature, its high sensitivity for very thin lms, its variable
information depth and the fast evolution of sophisticated equipment in recent years have
lead to the wide-spread application of Raman spectroscopy. The so-called micro-Raman
spectroscopy, which allows measurements of lateral resolution in the micro meter range,
has further widened the eld of applications.
In principle, the method is based on inelastic scattering of photons at lattice vibrational
modes, e.g. at phonons. By evaluating the phonon frequencies, and the intensities, half
widths, and shapes of lines in the Raman spectra the specic lattice dynamics of the sample can be investigated. This information can than be related to structural properties such
as identication of materials and compounds, composition of mixed compounds, layer orientation, stress and crystalline perfection. In addition resonance Raman spectroscopy can
be used to investigate the critical points in the electronic energy bands. Section 2.2.2.1 will
rst outline some fundamentals of Raman scattering. Then dominant lattice vibrational
modes in Cu-chalcopyrites will be presented in section 2.2.2.2. Subsequently, the deter-
2.2. Structural Characterization
29
mination of composition of mixed compounds, such as Cu(In1−x Gax )S2 , will be discussed.
Finally, a short description of the experimental set up is given in section 2.2.2.3.
2.2.2.1 Principles of Raman
Raman spectroscopy observes inelastic light scattering processes where energy of an incident photon of energy
~ωi
dierent energy
The energy transfered to to the sample corresponds to the energy
~Ωj
~ωs .
j.
of, e.g. a phonon
is transfered to the sample and to a scattered photon of slightly
Since the frequency of the incident light
by using a laser the energy
Ωj
ωi
can be well dened
of the phonon involved in the scattering process can be
obtained by analyzing the peak frequencies
ωs
of the scattered light. Energy as well as
quasi-momentum have to be conserved in the process:
~ωs = ~ωi ± ~Ωj ; Ks = Ki ± qj ,
where
The
Ki,s
± sign
refers to the photon wave vectors and
(2.1)
q
j to the wave vector of the phonon.
in Equation (2.1) results from the fact that in the scattering process a phonon
is either generated (Stokes process) of annihilated (Anti-Stokes process). In most experiments only Stokes processes are investigated. As a result of the conversation of energy
and momentum only certain combinations of energy and momentum can be transfered to
6
−1
the sample. Typical | j | values lie in the range of 10 cm , hence only phonons in the
q
immediate vicinity of the center of the Brillouin zone can be involved in a Raman process.
In the scattering process the interaction between the photon and the phonon acts via electronic inter-band transitions (indirect process). According to [74] Raman light scattering
can be described in a microscopic quantum mechanical time-dependent perturbation theory.
Here the photon-electron and the electron-lattice interaction is described by three
electronic transitions:
1- the absorption of a photon leading to an electronic transition from a ground state
to an excited state
|e.
2- the electron-lattice interaction, i.e. the electronic transition from a state
|e which involves the creation or annihilation of a photon Ω.
~
|e
|0
to a state
~
3- the recombination of the electron-hole pair under emission of a photon ωs which cor
responds to the transition |e to |0.
In this picture the transition |e → |e corresponds to transitions between intermediate states in the same band caused by a phonon-induced energy shift of the band itself
(intra-band electron-phonon interaction). Inter-band electron-phonon interactions are also
possible, however the scattering eciency is much smaller. Since the energy conservation
needs only to be full-lled for the process as a whole the electron-hole pairs states involved
in the process are virtual states and must not necessarily coincide with the real electronic
band structure.
If, however, transition energies correspond to real states in the band
structure the transition probabilities drastically increase leading to much higher Raman
scattering cross sections for excitation energies close to e.g. band gap energies (resonant
Raman scattering).
30
Experimental Procedure
2.2.2.2 Raman at Cu-chalcopyrites
One of the rst reports of Raman spectroscopy studies at
a chalcopyrite semiconductor (ZiSiP2) was published by
Kaminow et al. [76]. Later Koschel and Bettini [77] presented lattice dynamical calculations of some chalcopyrite compounds and were able to predict some so far
unobserved modes.
A detailed report on experimental
Raman Intensity
results of the Raman spectra of CuInSe2 was published
by Tanino et al. [78]. Meanwhile there is a considerable
number of publications regarding Raman spectroscopy of
Cu-chalcopyrites single crystal and thin lm specimen. A
concise review covering experimental results as well as lattice dynamical calculation of chalcopyrite compounds has
been given recently in a series of publications by Ohrendorf and Haeuseler [7981].
The chalcopyrite-structure has four formula units in the
tetragonal unit cell. The primitive cell contains two for-
0
20
40
60
Photon Energy (meV)
mula units, i.e. eight atoms per unit cell. Therefore 24 vibrational modes (3 acoustic, 21 optical) can be expected.
In order to detect a phonon in a Raman experiment the
Posi-
phonon must obey certain necessary conditions (Selection
mode as
Rules), which are based on symmetry considerations de-
a function of composition in
rived form the crystal structure under investigation. If the
Figure
2.6:
Spectral
tion of Raman A1
Cu(In1−x Gax )S2 ,
x
x
= 0.86 (b),
= 0.3 (d),
x
x
x
= 1 (a),
= 0.5 (c),
= 0 (e) [75].
phonon can be observed in Raman it is also called Ramanactive.
According to a group theoretical treatment by
Kaminow et.al. [76] the following irreducible representation of the optical phonons at the zone center results
Γ = A1 (Ra) + A2 + 3B1 (Ra) + 3B2 (IR, Ra) + 6E(IR, Ra) .
Ra
and
IR
in parentheses stands for Raman and Infra-red active respectively.
(2.2)
In the
case of non-resonant scattering the Raman component corresponding to the A1 mode is
much stronger than any other. Therefore the analysis of Raman spectra of CuInS2 and
Cu(In1−x Gax )S2 thin lms will be based mainly on the A1 phonon line. The A1 mode is
caused by the vibration of two pairs of anions, with one vibrating in the direction of the
a-axis
and the other in the direction of the
b-axis.
Since optical phonons are a fairly unique ngerprint of a certain material, phase identication by Raman is a more or less straightforward procedure.
Furthermore variations
of the composition of a compound can well be resolved from the energies of the phonon
modes. The relationship between the vibrational dynamics of a mixed compound and its
composition (or stoichiometry) can be classied into three types: i.e. one-mode, two-mode,
2.2. Structural Characterization
31
and three-mode behavior. In the one-mode case the constituent compounds have the same
phonon modes and their frequencies varies approximately linearly with composition between the respective end point values. In the two-mode case, the intermixing materials
exhibit separate phonon modes, which vary characteristically, in general not linearly, with
composition. At intermediate composition modes of both materials will be observed. In
the three-mode case additional modes, only characteristic for the mixed compound, appear
beside the modes of the endpoint materials.
The vibrational system of solid solutions in Cu-chalcopyrites transforms in the two-mode
manner in the case of anion substitution and in the one-mode manner in the case of
cation substitution [75, 82]. Correspondingly the CuInS2 -CuGaS2 system shows one-mode
or uni-modal behavior.
This was experimentally conrmed by Agkyan et al. [75] who
found an continuous shift in the A1 peak position with composition
found that in the range of 0.3 <
x
x.
Furthermore they
< 0.8 the mode was strongly broadened (Figure 2.6).
As outlined above the A1 is caused by the vibrational motion of the anion lattice. The
shift of its spectral position is mainly due to the slightly altered Ga-S binding force with
respect to In-S and the smaller atomic weight of Ga. The shift between CuInS2 and
−1
CuGaS2 is about 18 cm
(36 meV to 39 meV in Figure 2.6). Taking into account the
−1
typical experimental resolution limit of 12 cm
the composition can be determined to
an accuracy of about 5 %. This might seem to be a moderate value compared to methods
such as XRD, which, in general, give results of much higher accuracy based on precise
lattice constant determinations. Nevertheless the determination of composition by means
of Raman oers a number of advantages.
Particularly in thin lm analysis an optical
method, such as Raman, might be more suitable as the required minimum layer thickness
is lower than for X-ray diraction.
Furthermore in the case of CuInS2 the penetration
depths of the laser light used for excitation is only 100 nm (1/20 of the typical absorber
layer thickness), a fact that oers the possibility of depth-resolved measurement, when the
Raman measurement is combined with ion-etching methods. In micro-Raman mode the
high lateral resolution allows to perform homogeneity studies of composition in the micro
meter range across a samples surface.
Besides the identication of phases or the determination of compositions Raman spectrometry provides valuable information on the degree of crystallinity of a sample. Real
crystal structures are always the subject of impurities and dislocations. These eects may
be responsible for the non-conversation of momentum in the scattering process which is
experimentally reected by a corresponding broadening of the spectral lines.
The line
shape of phonon peaks observed in a Raman spectrum can thus be used to characterize
the degree of disorder in a solid. The lifting of momentum conservation with structural
disorder can even lead to acoustical phonons contributing to the Raman signal which will
lead to a very broad background in the spectra in the vicinity of the excitation laser line.
32
Experimental Procedure
2.2.2.3 Raman - Experimental
All micro-Raman spectra reported in this work were measured at the University of Barcelona.
The spectra were taken by means of a Jobin-Yvon T64000 spectrometer coupled with an
Olympus metallographic microscope.
submicronic (objective:
The obtained spot size on the sample was slightly
x100, NA = 0.95).
All spectra were recorded in backscattering
conguration. The sampling volume of a Raman measurement is limited on one hand by
the spot size on the other hand by the penetration depth of the laser light used for excita+
tion. In our case the green line of an Ar laser (λ = 514.5 nm) was used which corresponds
to a sampling depth of about 100 nm in CuInS2 ; less than one twentieth of the usual total
absorber thickness.
Depth-dependent information on lm structure could be gained by repeatedly sputtering
+
(Ar ) the sample and performing Raman measurements in between. A PHI 670 Scanning
Auger Nanoprobe System was used for sputtering.
was determined by
Auger Electron Spectroscopy
Simultaneously the atomic composition
(AES) between successive sputter steps.
A 10 nA current of 10 keV electrons was used for AES. The ion energy for sputtering
was 2 keV. The ion current was xed at a nominal value of 50 µA for all sputter steps.
Reference Raman measurements at CuInS2 single crystals did not reveal any signicant
sputter related damage under the described experimental conditions.
2.2.3 X-ray Diraction
XRD measurements served as a standard tool for phase identication throughout this work.
The method was also employed in order to resolve Ga-induced eects on structural parameters such as lattice constants, anion displacement, and tetragonal distortion by analyzing
the peak position and integral intensity of selective reections. XRD measurements were
performed using a Brucker D8 diractometer. This section starts with a brief overview of
fundamental aspects which determine the intensity of a X-ray reection. The inuence of
instrumental parameters as well as sample related parameters onto line shape and position
of a XRD reection will be briey discussed. Then numerical calculations of XRD-spectra
of CuInS2 thin lm samples will be outlined. Furthermore XRD-spectra of lms with gradients in lattice constant will be discussed. Correction factors necessary when analyzing
thin lm samples are discussed in Appendix A.3.
2.2.3.1 The scattered intensity
The scattered intensity in a X-ray diraction experiment can well be described in terms of
classical theory of scattering of X-rays by electrons (kinematical theory). When electromagnetic radiation falls on an atom, or more precisely on the electronic sphere of the atom,
two processes may occur: (1) the radiation may be absorbed with an ejection of electrons
2.2. Structural Characterization
33
from the atom, or (2) the radiation may be scattered. In X-ray diraction only the latter
process is important.
The scattered radiation, which spreads out in all directions from
the atom has the same frequency as the primary radiation (elastic scattering).
In high
resolution X-ray diraction the kinematical approach, which treats the electro magnetic
eld strength as a planar wave, fails to predict the experimentally observed intensities
and a more fundamental, dynamical theory, becomes necessary. The following outline is
restricted to the kinematical case.
In X-ray diraction the scattered intensity is caused by interference between waves scattered from the dierent atomic sites within the crystal.
When treating the scattering
classical reection of X-rays at a set of periodically spaced lattice planes the
common Bragg Law can be derived, which in vector form is given by
process as a
where
s − s0
= Ghkl ,
λ
2 sin Θ
1
s − s0
|=
.
|Ghkl | =
, |
dhkl
λ
λ
(2.3)
s0 and s are unit vectors in the directions of the primary and diracted beams k0
and k, Ghkl is called the scattering vector, dhkl is the distance between equivalent lattice
planes, and Θ denotes the angle between the lattice planes and the incident or the diracted
beam. The indices (hkl) are the Miller indices of the set of lattice planes which causes the
reection. In the case of an orthogonal crystal structure with lattice constants a, b, and c
the lattice plane spacing dhkl is given by
Here
1
d2hkl
=
h2 k 2
l2
+
+
.
a2
b2
c2
(2.4)
The Bragg-law describes the angle under which a XRD-reection will occur in experiment.
As mentioned above the scattered intensity is caused by interference between waves scattered from the dierent atomic sites within the unit cell. The total scattered intensity is
then obtained by 1.- summing over the scattered intensity of each atom
unit cell, usually described in terms of a atomic scattering factor
factor which accounts for the dierent positions
all unit cells
N
in the crystal
fα , multiplied by
a phase
in the unit cell and 2.- summing over
in the sample,
I(Ghkl ) ∝ K
Fhkl
rα
α
I0 2
N |Fhkl |2
r2
is called structure factor.
I0
where,
Fhkl =
fα e-i Ghkl ·rα .
(2.5)
r
for the distance
α
stands for the initial intensity and
between sample and detector. The factor
K
has been introduced to account for corrections
due to sample geometry and experimental set up. Several contribution to the correction
factor
K
are discussed in Appendix A.3. Equation (2.5) shows that all sample dependent
information regarding the intensity of a reection
(hkl)
is contained in
other terms in the equation are sample independent. The atomic position
Fhkl ,
whereas all
rα in the unit cell
34
Experimental Procedure
is usually expressed in terms of components along the basis vectors of the lattice
by means of fractional coordinates
written
Fhkl =
u, v, w ∈ [0 . . . 1].
a1 , a2 , a3
The structure factor may then be
fα exp[−2πi(huα + kvα + lwα )] .
(2.6)
α
In brief, the shape and dimension of a crystal unit cell can be deduced from the position
of XRD-reections, the content of the cell can be determined from the intensity.
2.2.3.2 Peak prole analysis
In an experiment the overall intensity of a XRD-reection will be spread out over a nite
angular range. The width of this range, i.e. the full width at half maximum (FWHM) of
the XRD reection, is determined by several independent eects. The nal line prole of
the reection can be considered as a convolution product of
•
the emission prole of the source
•
specimen aberrations due to nite grain size, strain, and defect concentrations, as
well as absorption in the sample, nite sample thickness, and tilt of the sample
•
an instrument component which determines the angular resolution of the experimental set up
With current diractometers instrument broadening can be minimized to a level, so that,
assuming a perfect sample, the emission prole becomes a signicant part of the total observed line prole. The emission prole of the Cu-Kα radiation mainly consists of two contributions Kα1 and Kα2 . Due to the ne-structure of the underlying electronic transitions
the emission lines are slightly asymmetric. The prole can be suciently approximated by
means of two Lorentz-proles [83]. Since the incident radiation is never exactly monochromatic the half width of a reection is broadened (dispersion). In XRD spectra collected
in this work the peak splitting due to the emission line doublet Kα1 and Kα2 could be well
resolved. However, the peak shapes were either determined by instrumental or by sample related broadening. Sample broadening arises mainly from the fact that the scattered
intensity is due to a coherent superposition of scattered X-ray waves. In practice only a
nite volume of the crystal will lead to coherent scattering. Eects which determine this
volume might be crystallographic defects (point defects, dislocations), grain boundaries,
or the nite thickness of the sample itself. According to Scherrer [84] the nite grain size
d
in a polycrystalline sample leads to a broadened intensity prole
∆|G| = 0.9 · 2π/d .
I(G)
of half width
(2.7)
The corresponding line prole can be described by a Lorentz-prole. The FWHMgs of such
a grain-size broadened prole was given by Cheary [85] as
FWHMgs
_ cos1Θd ,
(2.8)
2.2. Structural Characterization
where
[d]
35
= nm. In the case of polycrystalline samples with grain sizes
range the above relation allows for a determination of
d
d in
the submicron
by evaluating the FWHM of the
line prole of the XRD reection, presuming instrumental broadening can be neglected or
has been corrected for.
Figure 2.7: The six instrumental weight functions for a (A) low-resolution and (B) high
resolution diractometer [86].
A detailed overview about eects leading to instrumental broadening can be found in [87].
Eects, which aect the shape, symmetry and position of a reection and which are introduced by the geometry of the experimental set up include [88]:
•
at specimen error
•
axial divergence or brush eect
•
specimen transparency
•
generator focus, eects of apertures, and misalignment
Each individual eect leads to a characteristic deviation of the peak prole. Figure 2.7
plots the experimental weight functions for the six most important experimental errors for a
low resolution and a high resolution diractometer [86]. A convolution of all eects results
in the nal prole . Among them the at specimen error and the axial divergence turned
out to be the most important ones in measurements of this work. The at specimen error
arises from the fact that a at sample lies tangential at the circle of focus (see Figure 2.9).
This leads to asymmetric broadening and a shift of the peak to lower
2Θ
values.
The
error can be reduced by means of apertures. The fact that the mean scattering volume is
located below the sample surface causes an additional peak shift due to deviations from
the ideal beam geometry. The magnitude of this shift strongly depends on the absorption
−1
coecient of the sample. However, for µ > 250 cm , as in CuInS2 , it turns out to be
◦
below ∆Θ = 0.01 [89]. The divergence of the incident and the diracted beam in the
36
Experimental Procedure
Brush eect.
plane normal to the circle of focus leads to the so-called
Numerically this
eect causes the largest asymmetrical broadening. It also leads to a Θ-dependent shift (to
◦
smaller values for Θ < 90 ). Again, the eect can be reduced by additional apertures in
the beam path (Soller-Blenden).
The focus of the primary beam generated in the X-ray tube is very sensitive to the operating
conditions of the tube.
Therefore scans were always performed at the same generator
voltage and current; the same values which were also used for system calibration. Changes
of the line shape due to the apertures in the beam path were negligible in our set up, when
compared to the eects discussed above.
Misalignment of the set up can lead to signicant shifts in the measured angular position of
the XRD reection. Here the biggest source of error will be introduced by a displacement
of the sample surface with respect to the ideal position of the circle of focus. As is depicted
in Figure 2.8 the peak shift is given by
2S cos Θ
= tan ∆2Θ ≈ ∆2Θ , Θ
R
(2.9)
R is the radius of the goniometer. The corresponding peak shift for an assumed misalignment of the sample surface of 50 µm are plotted in
Figure 2.8. In the range of low Θ values sample displacement was found to be the biggest
where
S
1,
is the sample displacement and
source for errors in peak position.
∆2Θ
R
s
0.03
symmetric scan
s = 50 µm
R = 217.5 mm
o
∆ 2θ ( )
2Θ
∆2Θmeas
0.02
asymmetric scan
Φ = 5.0
0.01
x = 2 S cos (Θ)
Φ = 0.5
o
o
0.00
Θ
Θ
x
s
Θ
20
40
60
80
o
2θ()
Figure 2.8: Systematic peak shift due to misalignment of sample.
In general measured patterns from an X-ray powder diractometer cannot be tted by a
simple analytical function such as a pure Gauss or Lorentz prole. Several analytical peak
2.2. Structural Characterization
37
shape functions had been proposed. Best results were obtained by using pseudo-Voigt or
Pearson-VII functions. These functions are not only characterized by peak position, intensity and full-width-at-half-maximum but also by an additional parameter
n
that denes
the fraction of Lorentzian to Gaussian character of the prole. Kern [88] has compared
several peak shape function for the precise determination of the peak position and obtained
best result when using a Pearson-VII function given by
Ii,k =
1/n
∆2Θi,k
2
Γ(n) (2 − 1)
1 + 4(21/n − 1)
Γ(n − 0.5)
π
F W HMk
F W HMk
2 −n
,
∆2Θi,k = 2Θi − 2Θk . Γ(n) is the gamma function. When n = 1
becomes a Lorentzian and when n → ∞ it becomes a Gaussian.
where
(2.10)
the Pearson-VII
2.2.3.3 XRD-Experimental
All XRD-scans where collected in Bragg-Bretano geometry (Figure 2.9). The Cu-Kα dou-
GM
G
DC
GS
DS
k0
D
G
FC
k
θ
θ
Figure 2.9: Bragg-Bretano geometry used in the Brucker D8 diractometer. G - generator,
D - detector, GM - Goebel Mirror, GS - generator Soller aperture, DS - detector Soller
aperture, FC - focusing circle, DC - detector circle.
blet from a Cu-anode (generator settings
E = 40 keV, I = 40 mA)
radiation. For better collimation of the incident beam a
the exit slit of the generator.
the incident beam.
was used as incident
Goebbel-mirror was
attached to
The Goebbel-mirror also removed Cu-Kβ radiation from
A szintilization counter was used as the detector.
To minimize ef-
fects due to spatial divergence of the reected beam a Soller-aperture (width=1 mm) was
38
Experimental Procedure
mounted in front of the detector entrance slit. The resolution limit of the diractometer
was checked using a reference quartz sample and a Al2 O3 (Corrund) powder. The Full◦
◦
Width-Half-Maximum (FWHM) of all peaks was in the range of 0.045 0.07 . Their was
no signicant angle dependence of the peak width; only a slight increase with decreasing
peak intensity. Two-theta osets were also checked using these standards.
Two scanning modes were used during this work, i.e. symmetric scanning and asymmetric
scanning. Standard symmetric XRD
Θ - 2Θ
scans give an integral picture of a thin lm
sample, since the intensity of the incident radiation is nearly homogeneous across the
sample thickness.
In order realize more surface sensitive scans asymmetric scans were
performed where the incident beam is xed at a very small angle
Φ
and only the detector
is scanned. Figure 2.10 gives a schematic overview of the two modes.
The penetration depth of the incident X-ray intensity into a thin lm, assuming an
k
G
k0
G
k
θ
θ
Ψ
Φ
(a)
θ +Ψ
k0
(b)
Figure 2.10: Experimental geometry of the Bragg-Bretano method for X-ray diraction;
(a) symmetric mode, (b) asymmetric mode.
absorption coecient of
µ = 730 cm−1
(corresponds to CuInS2 ), is plotted in Figure 2.11
for dierent angles of incidence of the primary beam. It can clearly be seen that for an
◦
incident angle of Φ = 5.0
the sampling volume accessible for XRD is of the order of
2 µm which is the typical thickness of samples investigated in this work. In the case of an
◦
angle of incidence of Φ = 0.5 the rst 500 nm only will contribute to the measurement.
Thus by varying
Φ
the surface sensitivity of the measurement can be inuenced. This is
particularly useful when analyzing bilayer structures of unknown stacking sequence. The
change in the intensity ratio of reections from the dierent layers with incident angle can
be used to determine the relative location of the dierent layers. X-ray absorption in a
thin lm sample as a function of incident angle is also discussed in Appendix A.3.
2.2. Structural Characterization
Rel. X-ray Intensity I / I0
1.0
39
-1
µCIS = 730 cm
labels denote
angle of incidence Φ
o
0.5
10.0
o
5.0
Figure 2.11: Calculated X-ray inten-
o
o
0.0
0.5
sity versus sample depth for dier-
o
0.9 1.8
surface
0.0
Mo back layer
0.5
1.0
1.5
2.0
Depth (µm)
ent angle of incidence of the primary
beam. The assumed X-ray absorp−1
cortion coecient of µ = 730 cm
responds to CuInS2 .
2.2.4 Calculation of XRD spectra
Calculated X-ray diraction spectra for Cu(In1−x Gax )S2 thin lms were employed in this
work in order to evaluate measured XRD data. A script based on the software package
(TM)
MATHEMATICA
[90] was written, so that eects of structural deviations and eects due
to gradients in lattice constant could be simulated.
Structure factors were calculated from Equation (2.6). Values for the atomic scattering
factors were interpolated form tabulated values taken from the International Tables of
Crystallography [91].
Dispersion was accounted for by applying the correction factors
of Cromer and Liberman [92].
Isotropic Debye-Waller temperature factors were taken
from the crystal renement of single crystals of CuInS2 and CuGaS2 by Abrahams and
Bernstein [23].
On the basis of the structure factors peak intensities were calculated
taking into account absorption of X-rays in the sample and the thin lm geometry. All
correction factors considered in the simulations are discussed in Appendix A.3. Calculated
peak proles were based on the Pearson-VII function according to Equation (2.10). The
full width at half maximum parameter (FWHM) and the peak-shape parameter
n
were
obtained from ts to experimental data of reference samples. All calculations were done
for Cu-Kα1 and Cu-Kα2 radiation with an assumed intensity ratio of Kα2 /Kα1 = 1/2. A
list of the parameters used can be found in Appendix A.4.
To compute the XRD-spectrum for a layer with variable lattice constant along the depth
prole, the layer was subdivided into laminae of constant lattice constants.
The XRD
spectrum was then obtained by summing up the reected intensities of each lamina corrected for absorption of the incident and the scattered beam in the overlying layers. The
minimum number of laminae required was determined empirically by recalculating the
spectrum with increasing number of laminae until no detectable change in the calculated
◦
output, with respect to the experimental resolution limit of 0.01 , occurred.
40
Experimental Procedure
2.2.4.1 X-ray structure factors of CuInS2 and CuGaS2
As shown in Section 2.2.3.1, Equation (2.6), the structure factor describes the coherent
superposition of scattered waves originating from the individual atomic positions in the
unit cell. Depending on the specic structure interference might be either constructive or
destructive, hence not all reciprocal lattice vectors which satisfy Bragg's law will lead to a
reection in an experiment.
For a chalcopyrite structures such as CuInS2 and CuGaS2 the allowed reections fall into
one of three categories [93].
Group (i) reections have
(h, k, l/2)
all even or odd. These reections correspond to the
zincblende structure. A I-III-VI2 compound with random occupation of the cation lattice
site (sphalerite) would lead to group (i) reections only.
the tetragonal unit cell is distorted, i.e.
2c = a.
In most ternary chalcopyrites
As a result the multiplicity of zincblende
h = l/2 or k = l/2 is reduced and the singular zincblende reection is split
(2, 0, 0)zincblende → (2, 0, 0)/(0, 0, 4)chalcopyrite .
Group (ii) reections have indices given by (h, k) even and (l/2) odd or vice versa. The
structure factor of these reections depends only on the anion scattering factor. If u = 0.25,
reections with
into a doublet, e.g.
i.e. no anion displacement, the structure factor vanishes.
Group (iii) reections have indices given by
(h)
even and
(k, l)
odd or
(k)
even and
(h, l)
odd. The structure factor contains a cation term which arises from the dierence in the
cation scattering factors and, in case
u = 0.25
also an anion term. These reections are
also called superlattice peaks.
A list of calculated
Fhkl
values for selected reections of CuInS2 and CuGaS2 can be found
in Table 2.1. The table also compares the calculated data to values reported by Jae and
Zunger [8]. Their values are based on the Fourier transform of the calculated electronic
charge densities, which were calculated self-consistently by means of a density-functional
formalism. Both sets of data were calculated with the same set of lattice parameters given
in [24] and listed in Table 1.1. The agreement is reasonably well, taking into account that
the calculations of this work are based on tabulated atomic scattering factors for neutral
atoms which do not account for changes in the electron density due to chemical bonding
and lattice formation. For a CuInS2 and CuGaS2 lm of 2.5 µm thickness the calculated
XRD intensity spectra (Θ-2Θ scan, Bragg-Bretano geometry) are depicted in Figure 2.12.
Using the classication (i)(iii) for chalcopyrite structure factors the following trends can
be deduced.
Group (i) or zincblende like reections are the most intense in the spectra. The (112) and
the (024)/(020) lattice planes have the highest packing density and therefore the corresponding peaks dominate the spectra. Due to the lower atomic scattering factor for Ga
compared to In, XRD reections for CuGaS2 are by one fth weaker in intensity than the
corresponding CuInS2 reections.
The eect of the tetragonal distortion of the chalcopyrite unit cell on the (024)/(220)
group (i) reections is demonstrated in Figure 2.13. The distortion
c/a is greater than 2 in
2.2. Structural Characterization
41
Table 2.1: Comparison of calculated structure factors
Fhkl
in
e/(primitive cell) of
selective
XRD reections of CuInS2 and CuGaS2 from Ref. [8] and this work.
CuGaS2
CuGaS2
CuInS2
CuInS2
hkl
group
[8]
this work
[8]
this work
(112)
(i)
112.77
113.28
142.63
143.35
(200)
(i)
53.32
53.87
84.94
85.70
(004)
(i)
51.98
52.57
82.78
83.44
(204)
(i)
123.04
124.60
151.57
153.37
(220)
(i)
122.86
124.41
149.43
151.18
(116)
(i)
86.21
87.64
112.28
113.74
(312)
(i)
86.43
87.86
111.01
112.42
(224)
(i)
45.67
46.16
73.50
73.74
(008)
(i)
102.12
103.61
129.07
130.74
(400)
(i)
100.70
102.20
122.42
123.98
(202)
(ii)
1.09
1.06
2.14
2.23
(310)
(ii)
1.56
1.77
3.28
3.64
(402)
(ii)
2.98
3.04
6.11
6.19
(101)
(iii)
6.63
6.61
19.56
19.72
(103)
(iii)
1.75
1.84
29.38
29.51
(121)
(iii)
6.85
6.81
29.11
29.09
(123)
(iii)
8.13
7.95
19.31
19.15
(233)
(iii)
6.83
7.08
30.30
30.23
CuInS2 and less than 2 in CuGaS2 . (Table 1.1). The values in the alloyed Cu(In1−x Gax )S2
system obey Vegard's law [25], i.e. they vary linearly with composition
x (Figure 2.13 (a)).
Figure 2.13 (b) shows calculated (024)/(220) reections of a Cu(In1−x Gax )S2 thin lm as
a function of composition
x.
The doublet structure with an intensity ratio
I220 /I024 ≈ 1/2
can clearly be seen. Although, the anion displacement changes quite signicantly between
CuInS2 and CuGaS2 its inuence on the structure factors of group (i) peaks is negligible.
Since the tetragonal distortion arises from the ordered cation sublattice in the chalcopyrite
structure the peak splitting serves as a good indicator for the degree of chalcopyrite cation
ordering in a sample. At
x ≈ 0.2
where
c/a = 2.0
the distortion vanishes and no peak
splitting is observed.
In such a case chalcopyrite ordering can only be evaluated by analyzing the group (iii)
peak intensities, which are determined by the dierence in scattering factor at the dierent
cation sites in the unit cell. In CuGaS2 the structure factors of this group are rather small,
i.e. below 10 e/cell (Table 2.1), since fCu almost equals fGa . As can be seen in Figure 2.12
◦
only the (101) reection at 18.0 can be expected to be above the detection limit. At
higher
2θ
values the superlattice of a CuGaS2 thin lm are very weak since the peak in-
tensities are signicantly reduced by the polarization and Lorentz-factors and in case of a
thin lm sample additionally by the reduced scattering volume. In CuInS2 the situation is
dierent as the large dierence between
fCu
and
fIn
leads to much higher structure factors
for group (iii) peaks. At least ve superlattice peaks can be well resolved at the simulated
(224)
(125)
(301)
(123)
0.1
(121)
(103)
(101)
0.2
(233)
(008)/(040)
(116)/(132)
zincblende peaks
superlattice peaks
(004)/(020)
CuInS2
(112)
0.3
(024)/(220)
Experimental Procedure
Norm. XRD Intensity
42
(220)/(024)
(233)
(301)
(123)
(121)
(020)/(004)
(103)
0.1
(132)/(116)
0.2
(224)
CuGaS2
(112)
0.3
(101)
Norm. XRD Intensity
0.0
0.0
o
20
o
30
o
40
o
50
o
60
o
70
2 Theta
Figure 2.12: Calculated XRD spectra of CuInS2 (upper plot) and CuGaS2 (lower plot).
Intensities are plotted relative to the (112) reection.
spectra of the CuInS2 thin lm in Figure 2.12. The inuence of structural deviations from
an ideal chalcopyrite unit cell onto the group (iii) reections of a CuInS2 thin lm such
as cation order/disorder transitions, deviations from molecularity, changes in the anion
displacement
u, and the isovalent substitution of In with Ga is discussed
in Appendix A.1.
Most of the group (ii) structure factors are well below 5 electrons/cell, which leads to reections that are in general below the detection limit of a XRD-measurement at a powder
sample. No group (ii) reection was observed at samples of this work.
2.2. Structural Characterization
43
CuGaS2
[Ga]/[In+Ga]
c Lattice
constant (nm)
1.12
1.08
0.54
1.06
1.04
c
1.02
2.1
2.0
distortion
0.52
0.9
0.8
XRD Intensity
1.10
c/a Tetragonal
constant (nm)
a Lattice
1.0
1.14
a
0.56
0.7
0.6
0.5
0.4
0.3
1.9
0.30
0.20
0.0
0.5
u Anion
displacement
0.25
0.2
0.1
CuInS2
o
1.0
46
[Ga]/[In+Ga]
o
47
0.0
o
48
o
49
2 Theta
(a)
(b)
Figure 2.13: (a) Lattice constants, tetragonal distortion and anion displacement of the
Cu(In1−x Gax )S2 alloy system as a function of
x and (b) corresponding XRD-spectra of the
(024)/(220) doublet for Cu-Kα1 and Cu-Kα2 radiation. The characteristic peak splitting
vanishes at [Ga]/([In] + [Ga]) ≈ 0.2 where
c/a = 2.0.